Presentation on the use of Slide Rule by Prof. Joseph Pasquaule

Transcription

The Slide Rule
Calculating by Mind and Hand
J. Pasquale
Joe Pasquale
Department of Computer Science and Engineering
University of California, San Diego
November 18, 2005
Oughtred Society Meeting, MIT
1
What is a Slide Rule?
Source: www.antiquark.c om/sliderule/sim
•
•
•
•
Analog calculator – by mind and hand
Scales on body and slide, with cursor
x×y, x÷y, 1/x, x2, 䌥㼛, x3, ҁ], xy, x1/y, …
10x, log x, ex, ln x, sin/tan, sinh/tanh, …
Joe Pasquale, UCSD
2
History
Napier
•
•
•
•
•
•
•
1614 Napier
1617 Briggs
1620 Gunter
1630 Oughtred
1850 Mannheim
1891 Cox
1972 HP
Joe Pasquale, UCSD
Oughtred
logarithms
common logarithms
logarithmic scale
slide rule
standardized scales
duplex slide rule
electronic calculator
3
Multiplication: 2 × 3
• 1/C above 2/D
• Cursor above 3/C
• Read 6/D
Joe Pasquale, UCSD
4
Multiplication: 2.15 × 3.35
• 1/C above 2.15/D
• Cursor above 3.35/C
• Read 7.20/D (why 7.20 and not 7.2?)
Joe Pasquale, UCSD
5
Multiplication: 76 × 0.32
• 1/C above 7.6/D – use right index of C
• Cursor above 3.2/C, read 2.4/D
• Correct for decimal point: 24
Joe Pasquale, UCSD
6
23.6
Source: web.mit.edu
Holman
Significant Digits
23.8
24.0
24.2
24.4
24.6
24.8
• 76 × .32 ≈ 24.32 [23.7825 – 24.8625]
– 75.5 × .315 = 23.7825
– 76.5 × .325 = 24.8625
• 76 × .32 ≈ 24
• 76 × .32 ≈ 24.3
Joe Pasquale, UCSD
[23.5 – 24.5 ] 2 SD
[24.25 – 24.35] 3 SD
7
Division: 5 ÷ 2
Source: http://www.antiquark.c om/sliderule/sim/
• 2/C above 5/D
• Read 2.5/D under 1/C
Joe Pasquale, UCSD
8
Division: 60 ÷ 0.24
Source: http://www.antiquark.c om/sliderule/sim
• 2.4/C above 6/D
• Read 2.5/D under 1/C
• Correct decimal point: 250
Joe Pasquale, UCSD
9
Algebra of Lengths
• Length(A + B) = Length(A) + Length(B)
A
B
A+B
• Length(A - B) = Length(A) - Length(B)
A
A-B
Joe Pasquale, UCSD
B
10
Calculating Power
• Any operation expressible in the form
A + B = C or A - B = C
can be implemented with a slide rule
• x × y = z 䊲 log x + log y = log z
• x ÷ y = z 䊲 log x - log y = log z
• xy = z
䊲 log log x + log y = log log z
Joe Pasquale, UCSD
11
Addition: Adding Lengths
1
0
2
0
1
2
1
2
3
3
3
• Example: 1 + 2 = 3
• L(1) + L(2) = L(1+2) = L(3)
Joe Pasquale, UCSD
12
Re-Label Scale Indices
0
logb b0
1
logb b1
2
logb b2
3
logb b3
• x = logb bx, for any x, for any b
• 0 = logb b0, 1 = logb b1, 2 = logb b2, …
Joe Pasquale, UCSD
13
Multiplication: Length 䍬㻃Log
1
logb b0
2
logb b0
logb b1
logb b2
logb b1
logb b2
logb b3
logb b3
3
• logbb1 + logbb2 = logbb3
• b1 × b 2 = b 3
Joe Pasquale, UCSD
14
Re-label Scale Indices
0
log2 20
1
log2 21
2
log2 22
3
log2 23
20
21
22
23
1
2
4
8
Joe Pasquale, UCSD
15
Add Intermediate Labels
1
2
4
8
1.000
1
2
3
4
5 6 7 8 91
1.585
• “x” is located at log x / log 2
• “3” is located at log 3 / log 2 ≈ 1.585
Joe Pasquale, UCSD
16
To Multiply, Add Exponents
1
2
1
1
2
2
3
4
3
4
5 6 7 8 91
5 6 7 8 91
3
• 21 × 22 = 21+2 = 23
• 2 × 4
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= 8
17
Multiplication and Division
1
1
2
2
3
4
3
4
5 6 7 8 91
5 6 7 8 91
• 2 × 2 = 4, 2 × 3 = 6, 2 × 4 = 8, 2 × 5 = 10
• 4 ÷ 2 = 2, 6 ÷ 3 = 2, 8 ÷ 4 = 2, 10 ÷ 5 = 2
Joe Pasquale, UCSD
18
Multipliers Shift Scales
1
π
2
4
3
5 6 7 8 91
4
5 6 7 8 91
2
3
• Multiplication by π, shift scale to left
• 2 × π ≈ 6.28
Joe Pasquale, UCSD
19
Reciprocals Invert Scales
1
2
19 8 7 6 5
3
4
4
3
5 6 7 8 91
2
1
• Reciprocal: scale inverted horizontally
• 1/2 = .5, 1/3 ≈ .33, 1/4 = .25, 1/5 = .2
Joe Pasquale, UCSD
20
Powers Compress Scales
1
1
2
3
2 3 4 5 67891
4
5 6 7 8 91
2 3 4 5 67891
• Square: compress scale by factor of 2
• 22 = 4, 32 = 9, 52 = 25, 82 = 64
Joe Pasquale, UCSD
21
Roots Expand Scales
1
1
2
3
4
5 6 7 8 91
2
4
• Square root: expand scale by factor of 2
• 䌥㻕 ≈ 1.41, 䌥㻗 = 2, 䌥㻜 = 3
Joe Pasquale, UCSD
22
Looking at a Real Slide Rule
K
A,B
CI
C,D
•
•
•
•
C, D
CI
A, B
K
reference scales
reciprocal of C – inversion
square of C, D – 2x compression
cube of C, D
– 3x compression
Joe Pasquale, UCSD
23
Precision
• Depends on physical length
• 10 inch rule: 3-4 digits
• Ways to increase precision
– Increase physical length
– Wrap scale around rule to increase length
– Magnify the area of focus
Joe Pasquale, UCSD
24
Precision — Relative Error
• Compare physical distances at extremes
– Distance (1.00, 1.01) ≈ Distance (9.9, 10)
– (1.01-1.00)/1.00 = 1%, (10-9.9)/10 = 1%
• Relative error uniform across log scale
Joe Pasquale, UCSD
25
Precision vs. Accuracy
• 2×3=6
– accurate, not precise
• 2.00 × 3.00 = 6.01
– more precise, less accurate
• Are 2 and 2.00 located at same place?
– Does it matter? Why?
Joe Pasquale, UCSD
26
Trigonometry
c
θ
b
a
• Recall sin θ = b/c, cos θ = a/c, tan θ = b/a
• Scales for sin θ and tan θ
• To calculate cos θ, use sin 90-θ
Joe Pasquale, UCSD
27
Sin and Tan Scale Ranges
• Sin scale: 5.74 – 90.0 degrees
– sin 5.74 ≈ 0.1, cos 84.26 ≈ 0.1
– sin 90 = 1.0, cos 0 = 1.0
• Tan scale: 5.71 – 45 – 84.3 degrees
– tan 5.71 ≈ 0.1
– tan 45 = 1.0
– tan 84.3 ≈ 10
Joe Pasquale, UCSD
28
sin θ ≈ tan θ, for small θ
c
θ
b
a
• sin θ = b/c, tan θ = b/a
• For small θ
– a ≈ c, therefore sin θ ≈ tan θ
– Use ST scale for θ < 5.74
Joe Pasquale, UCSD
29
Calculating Arbitrary Powers xy
• xy can be calculated as A + B = C
xy 䊲 log xy = y log x
䊲 log log xy = log y + log log x
䊲 log log x + log y = log log xy
A
B
C
• Note that A and C are same scales: LL
• LL scales devised by Roget in 1815
Joe Pasquale, UCSD
30
ln 1+x ≈ x for small x
1
• Near x = 1, ln 1+x ≈ x (linear)
• log 1 = 0
Joe Pasquale, UCSD
31
How Were Scales Built?
• The Gilligan’s Island Slide Rule Problem
– You are stranded on an island
– You, “the professor,” must save the crew
– You decide to build a slide rule
• How do you determine graduations for …
– a log scale, log log scale, sin scale, tan scale
• Arithmetic + geometry, no calculators
Joe Pasquale, UCSD
32
Slide Rule Topology
• Slide rules come in many
– physical shapes and sizes
– scale configurations, lengths, layout
• Precision
• Size
• Convenience
Joe Pasquale, UCSD
33
Linear
K&E Deci-Lon
Pickett N3
Post Versalog
Faber Castell 2/83N
Source: www.sphere.bc.ca
Joe Pasquale, UCSD
34
Faber Castell 8/10
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Circular
Pickett 110ES
Source: www.sphere.ba.ca
35
Atlas
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Source: www.scienceandsociety.co.uk
Source: www.hpmuseum.org
Spiral
Sutton (1663)
36
Cylindrical Spiral
Source: www.hpmuseum.org
Stanley Fuller
Joe Pasquale, UCSD
37
Cylindrical Grid
Source: www.oughtred.org
Thacher 4012
Thacher 4013
Source: www.daviscenter.wit.edu
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38
Complex Arithmetic
Source: www.rechenwerkseug.de
Stanley Whythe
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39
Dimensional Analysis
K&E Analog
Source: www.mccoys-kecatalogs.com
Joe Pasquale, UCSD
40
UCSD Freshman Seminar
Masters of the Slide Rule, Winter ‘03
Joe Pasquale, UCSD
41
What Students Learn
•
•
•
•
•
How to use all scales
Estimation
Approximation
Precision, accuracy
Advanced topics
– Scales from scratch
– Benford’s Law
Joe Pasquale, UCSD
42
Larger Lessons
• Economy of calculating
– slide rules
– calculators
– computers
• Social value
– parents, grandparents
– do so much with so little
Joe Pasquale, UCSD
43
Quotes
My skills of estimation are getting
better … I like being engrossed in
the calculations, instead of just
punching them into my calculator. I
make less mistakes, and find I know
what I am talking about …
- Brian Robbins, W03
Joe Pasquale, UCSD
44
Quotes
I was looking at the A scale and I
liked how it finds squares by just
decreasing the size of the D scale
by half … So then I found the cubed
K scale, and of course, it is three
times smaller than the D scale.
- Tracy Becker, W03
Joe Pasquale, UCSD
45
Quotes
I like being able to see mathematical
operations in the visual way that a
slide rule allows … This seminar has
given me a better understanding of
precision, relationship between logs
and multiplication, and Benford’s Law.
- Amy Cunningham, W03
Joe Pasquale, UCSD
46
Quotes
What amazes me the most about
the slide rule is that it works … I
can’t help but marvel at its design
and that someone actually was able
to create such a device ... Its
complexity is just mind boggling.
- Kendra Kadas, F03
Joe Pasquale, UCSD
47
Quotes
I was in physics class, and the
professor explained how tan and sin
are close for really small angles. The
class didn’t show much reaction, but
my first thought was “hey, I learned
that from my slide rule seminar.”
- John Beckfield, F03
Joe Pasquale, UCSD
48
Quotes
The first couple of days with this
slide rule have really been a learning
experience for me … It took me
some time to realize that you could
multiply by any interval of 10 using
the same number spectrum.
- Rajiv Rao, F04
Joe Pasquale, UCSD
49
Quotes
This slide rule seminar is the only
thing saving me from a quarter full of
literature writing, and other
humanitarian monotony. After hours
of “theory of literature,” I realized I
still had slide rule homework. Hurray!
- Lydia McNabb, F04
Joe Pasquale, UCSD
50
Quotes
The slide rule rules. The slide rule is
truly an extension of a person, not
something completely separate such
as the calculator. I actually had to
think before, during, and after
getting the answer on the slide rule.
- Lynn Greiner, F04
Joe Pasquale, UCSD
51
Quotes
I’m actually quite amazed with the
design of the slide rule. I find the
folded scales especially ingenious
… I definitely feel I understand what
I’m doing - not quite the “black box”
that calculators are.
- Ryan Lue, F04
Joe Pasquale, UCSD
52
Quotes
The more I use the slide rule, the
greater the insight I have into how
ingeniously the scales were put
together. I hope I can re-teach my
parents how to use it.
- Chris Brumbaugh, F04
Joe Pasquale, UCSD
53
Proof of Slide Rule Use in ‘76
Student shows teacher a slide rule calculation.
Weehawken High School, NJ, 1976
Joe Pasquale, UCSD
54
Are We Making Progress?
Joe Pasquale, UCSD
55
FOR MORE INFO
Joe Pasquale
Dept. of Computer Science & Engineering
University of California, San Diego
9500 Gilman Drive
La Jolla, CA 92093-0404
[email protected]
http://www-cse.ucsd.edu/~pasquale
Joe Pasquale, UCSD
56
Supplemental
Joe Pasquale, UCSD
57
Optimal Length of Log Scale
• What integer total length L minimizes
RMS error of integer tick mark values?
• Determine for each tick mark X
– round (L * log(X))
• Compute Error
– | true value - nearest integer value |
• RMS: Root Mean Square (of errors)
Joe Pasquale, UCSD
58
Survey of Best Values < 1000
Length Error
63
10.86
176
9.99
239
7.89
329
5.90
392 10.24
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Length
505
568
744
807
897
Error
9.52
2.19
10.22
9.93
4.16
59
Length of 568, 2.2% error
Location of major tick marks
1:
2:
3:
4:
5:
0
171
271
342
397
Joe Pasquale, UCSD
0.00
170.99
271.01
341.97
397.02
6:
7:
8:
9:
1:
442
480
513
542
568
441.99
480.02
512.96
542.01
568.00
60
1:
0
0.00
2: 99
99.04
3: 157 156.97
4: 198 198.08
5: 230 229.96
Joe Pasquale, UCSD
6:
7:
8:
9:
1:
256
278
297
314
329
256.01
278.04
297.12
313.95
329.00
61
1:
2:
3:
4:
5:
0
118
187
236
274
Joe Pasquale, UCSD
0.00
118.00
187.03
236.01
274.00
6:
7:
8:
9:
1:
305
331
354
374
392
305.04
331.28
354.01
374.06
392.00
62

Presentation on the use of Slide Rule by Prof. Joseph Pasquaule

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